\paragraph{Interpass Durations}\label{par:analysis-of-interpass-durations-in-the-reversing-mill}

\begin{figure}[t]
    \centering
    \includegraphics[width=\linewidth]{img/hist_pauses}
    \caption{Histograms of Interpass Duration Between Roll Passes of the Reversing Mill}
    \label{fig:hist_pauses}
\end{figure}

For the analysis of the interpass durations a set of 29 rolling trials was analysed.
The histograms of the obtained dataset are shown in \autoref{fig:hist_pauses} in conjunction with the fitted distribution parameters.
The mean interpass duration was about $\Estimated{\Expectation}(\InterpassDuration) = \qty{6.1}{\second}$.
The \qty{90}{\percent} confidence interval is $[\qty{4.50}{\second}, \qty{9.32}{\second}]$.
Data points larger than \qty{15}{\second} were capped as outliers, since they occurred very rarely and are  related to operational problems.

\paragraph{Simulation Showcases}\label{par:simulation-showcases}

\begin{figure}[b]
    \hfill
    \centering
    \begin{subfigure}{0.5\linewidth}
        \centering
        \includegraphics[width=\linewidth]{img/sim1_temperature}
        \caption{Variation of the Initial Workpiece}
        \label{fig:sim1_temperature}
    \end{subfigure}%
    \begin{subfigure}{0.5\linewidth}
        \centering
        \includegraphics[width=\linewidth]{img/sim2_temperature}
        \caption{Variation of the Initial Workpiece and of the Interpass Durations}
        \label{fig:sim2_temperature}
    \end{subfigure}

    \caption{Temperature Evolution Within the Rolling Process Under Two Variation Assumptions}
    \label{fig:sim_temperature}
\end{figure}

\begin{figure}[t]
    \centering
    \includegraphics[width=0.5\linewidth]{img/temperature_std}
    \caption{Evolution of the Temperature Standard Deviation}
    \label{fig:temperature_std}
\end{figure}

In the following, two showcases of the Monte Carlo approach described above shall be shown.
For each case 100 sample conditions were drawn using a random number generator.
The current analysis was limited to the aspect of temperature evolution as example for the possibilities of the method, although other key problems such as roll force and torque, spreading and filling, as well as elastic rolling stand behavior are included in the simulation core.
The temperature evolution was selected, because it is the most vivid property of the process to show the importance of variation analysis.
It influences directly and substantially the plastic material behavior and the evolution of the microstructure, while it is quite sensitive to most process conditions.
More detailed data is included in the supplemental material~\cite{esaform2023_weiner_supp}.

The first showcase is the analysis of variations solely in the input workpiece.
Exemplary, the diameter $\Diameter$ of the round profile and the mean initial temperature $\Temperature$ of the workpiece were chosen.
Their variations were modelled using normal distributions with the nominal values as mean $\Expectation(\Diameter)=\qty{50.0}{\milli\meter}$ resp. $\Expectation(\Temperature)=\qty{1373}{\kelvin}$ and standard deviations of $\StandardDeviation(\Diameter)=\qty{1.0}{\milli\meter}$ resp. $\StandardDeviation(\Temperature)=\qty{10}{\kelvin}$.
The interpass durations were here chosen as an approximate constant of $\InterpassDuration = \qty{6.1}{\second}$ (corresponding to the mean interpass duration above) for the reversing passes, but were calculated directly from rolling velocity and distance for the continuous passes.

The resulting temperature evolution is shown in \autoref{fig:sim1_temperature}.
The variations of the incoming and outgoing temperatures at each roll pass are plotted as boxplots in the respective positions.
Two line plots show the results of the simulation using only the nominal values and the mean curve of the Monte Carlo results.
These were rather equivalent in this case.
The variations of the simulation results tended to decrease along the pass schedule, compare also \autoref{fig:temperature_std}.
In the continuous passes, the variations of the workpiece temperatures were rarely significant.
This is in accordance to practical experience.

The second showcase is the consideration of the varying inter-pass durations during operation of the reversing mill due to manual feeding.
The durations were drawn from the gamma distribution created above.
The remaining parameters were the same as in the first case.

The resulting temperature distribution is shown in \autoref{fig:sim2_temperature} similarly to the first case.
The standard deviations of both cases are compared in \autoref{fig:temperature_std}.
In contrast to the first case, the variations of the workpiece temperature are not decreasing along the pass schedule but in the finishing passes, since additional variation is introduced between each pass of the reversing mill.
So the variation of the output workpiece is higher than that of the input.
Additionally, the mean of the Monte Carlo results tends to deviate from the nominal results, unlike the first case.